p-Adic Aspects of Modular Forms

The following sections are included:

• Preface
• Contents

The arithmetic theory of modular forms has two main themes that are intertwined. One is the theory of congruences of modular forms and the other is the theory of Galois representations. The first theme is classical, dating back to Ramanujan, with subsequent important contributions from Serre, Swinnerton-Dyer, Atkin, Ribet, Hida etc, and the developments in the latter theme started with the work of Deligne, Eichler, Shimura, Serre and others. It opened up new frontiers in the last few decades of the twentieth century with work of Hida, Mazur, Taylor, Wiles, etc. This expository article is based on a series of lectures given at IISER Pune in June 2014, in the ‘Workshop on p-adic aspects of modular forms’. The subject of the lectures was the classical approach of Serre [Se1] in defining p-adic families of modular forms. Of course, the geometric approach as developed by Katz, Dwork, Coleman, Mazur, culminating in the theory of overconvergent modular forms is central to the study of p-adic modular forms today, but we shall not discuss this and refer the interested reader to [K1], [K-M].

In these lectures, we present, to some extent, the theory of ordinary p-adic families of Siegel cusp forms of arbitrary genus g, by using the geometry of the ordinary locus of the Siegel varieties. One part consists in studying the ordinary part of the space of Katz p-adic Siegel (cuspidal) modular form. Another is to relate this ordinary part to the ordinary part of spaces of classical (cuspidal) Siegel modular forms; this is the question of control (a terminology due to Mazur in the context of Iwasawa theory). The two parts are actually intertwined. Prominent in the first part is a p-level descent lemma due to Hida. In both, the “contraction lemma” (also due to Hida), which allows to compare algebraic and topological differential forms, plays a crucial role. For the question of control, the key object is the Hodge-Tate-Igusa map; it maps the Igusa tower into the torsor of bases of differentials over the ordinary locus. The behaviour of the pull-back by this map of ordinary Siegel (cusp) forms has been clarified by V. Pilloni when he introduced an intermediate tower between the Igusa and the torsor of bases of differentials (denoted in Sect. below). The outcome of this is the construction of the ordinary part of the eigenvariety for overconvergent Siegel cuspforms. Actually, a variant of this intermediate tower is a crucial tool in the construction of the full eigenvariety for overconvergent Siegel cuspforms (see [AIP12]). This method of construction is originally due to Hida [Hi02], who developed his ideas in the framework introduced by Katz [Ka73]; Hida's results have been improved by V. Pilloni [Pi12a], whose presentation we mostly follow. The construction of the full eigenvariety generalizing this method has been done first by [AIP12]; another construction of the eigenvariety along similar ideas has been done by [BMT14]. Note that, as soon as g > 1, the study of the blowing-down map from a toroidal compactification to the minimal compactification (restricted to the ordinary locus) plays a crucial role in several steps of the construction. By lack of time, we'll mention where but be very brief on details.

Let f be a modular eigenform with Fourier coefficients a_n. A p-adic family passing through f is a deformation of the Fourier coefficients a_n in a p-adic analytic way. By evaluating these p-adic analytic functions at certain weights, one gets the Fourier coefficients of other modular forms that are p-adically ‘close’ to f. Historically, the first example of such a family is the family of Eisenstein series. All the classical Eisenstein series G_k as k varies over even integers congruent to 0 modulo p−1 live in the same p-adic family (more precisely, this is true after removing some Euler factors from the Eisenstein series). The existence of such a family crucially relies on the construction of the p-adic zeta function…

This is an expanded version of a series of lectures given by the author at the workshop ‘p-adic Aspects of Modular Forms’ held in IISER, Pune in June 2014. The goal of the lectures was to explain modularity lifting and the Taylor–Wiles method in the specific case of ordinary n-dimensional Galois representations.

The use of modular symbols to attach p-adic L-functions to Hecke eigenforms goes back to the work of Manin et al in the 70s. In the 90s Stevens proposed a new approach based on his theory of overconvergent modular symbols, which was successfully used to construct p-adic L-functions on the eigencurve for GL₂ over ℚ. Recently, building on Urban's construction of eigenvarieties for general reductive groups and on the author's theory of automorphic symbols for GL₂ over a totally real number field, Barrera gave a new construction of p-adic L-functions for Hilbert modular forms using the overconvergent compactly supported cohomology of Hilbert modular varieties.

In addition to giving an overview of these topics, the lecture notes also contain some original results such as the precise correspondence between automorphic and modular symbols for GL₂ over totally real number fields.

The following sections are included:

• Introduction
• Some ring theory
  • Differentials
  • Congruence and differential modules
• Deformation rings
  • One dimensional case
  • Congruence modules for group algebras
  • Proof of Tate's theorem
  • Two dimensional cases
  • Adjoint Selmer groups
  • Selmer groups and differentials
• Hecke algebras
  • Finite level
  • Ordinary of level Np^∞
  • Modular Galois representation
  • Hecke algebra is universal
• Analytic and topological methods
  • Analyticity of adjoint L-functions
  • Integrality of adjoint L-values
  • Congruence and adjoint L-values
  • Adjoint non-abelian class number formula
• p-Adic adjoint L-functions
• References

These are the expanded notes of a mini-course of four lectures by the same title given in the workshop “p-adic aspects of modular forms” held at the IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GL_n with the specific aim to understand the p-adic symmetric cube L-function attached to cusp forms on GL₂ over the rational numbers.

Generalised Heegner cycles are associated to a pair of a normalised newform and a Hecke character over an imaginary quadratic extension. We survey the non-triviality of generalised Heegner cycles over anticyclotomic extensions of the imaginary quadratic field.

The purpose of this short note is to introduce the explicit reciprocity law of Heegner points in my joint work with Francesc Castella and explain some arithmetic applications. The text is based on my talk in ”p-adic aspects of modular forms” held in IISER, Pune. The details can be found in [1].

In this short note we partially answer a question of Fukaya and Kato by constructing a q-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.